1 Introduction

In this paper, we are concerned with wave equations associated with some Liouville-type problems on compact surfaces arising in mathematical physics: sinh-Gordon equation (1.1) and some general Toda systems (1.7). The first wave equation we consider is

$$\begin{aligned} \partial _t^2u-\Delta _gu=\rho _1\left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-u}}{\int _{M}e^{-u}}-\frac{1}{|M|}\right) \quad {\text{ on} }\,\, M, \end{aligned}$$
(1.1)

with \(u:{\mathbb {R}}^+\times M\rightarrow {\mathbb {R}}\), where (M, g) is a compact Riemann surface with total area |M| and metric g, \(\Delta _g\) is the Laplace–Beltrami operator, and \(\rho _1, \rho _2\) are two real parameters. Nonlinear evolution equations have been extensively studied in the literature due to their many applications in physics, biology, chemistry, geometry and so on. In particular, the sinh-Gordon model (1.1) has been applied to a wide class of mathematical physics problems such as quantum field theories, non-commutative field theories, fluid dynamics, kink dynamics, solid-state physics, nonlinear optics and we refer to [1, 8, 12, 14, 36, 37, 47, 51] and the references therein.

The stationary equation related to (1.1) is the following sinh-Gordon equation:

$$\begin{aligned} -\Delta _gu=\rho _1\left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-u}}{\int _{M}e^{-u}}-\frac{1}{|M|}\right) . \end{aligned}$$
(1.2)

In mathematical physics, the latter equation describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices, see [29, 41]. For more discussions concerning the physical background we refer, for example, to [13, 33, 35, 38, 39] and the references therein. On the other hand, the case \(\rho _{1}=\rho _{2}\) has a close relationship with constant mean curvature surfaces, see [48, 49].

Observe that for \(\rho _{2}=0\), Eq. (1.2) reduces to the following well-known mean field equation:

$$\begin{aligned} -\Delta _g u=\rho \left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) , \end{aligned}$$
(1.3)

which has been extensively studied in the literature since it is related to the prescribed Gaussian curvature problem [4, 43] and Euler flows [9, 30]. There are by now many results concerning (1.3), and we refer to the survey [45]. On the other hand, the wave equation associated with (1.3) for \(M={\mathbb {S}}^2\), that is

$$\begin{aligned} \partial _t^2u-\Delta _g u=\rho \left( \frac{e^u}{\int _{{\mathbb {S}}^2}e^u}-\frac{1}{4\pi }\right) \quad {\text{ on }}\,\, {\mathbb {S}}^2, \end{aligned}$$
(1.4)

was recently considered in [11], where the authors obtained some existence results and a first blow-up criterion. Let us focus for a moment on the blow-up analysis. They showed that in the critical case \(\rho =8\pi\) for the finite time blow-up solutions to (1.4), there exist a sequence \(t_k\rightarrow T_0^{-}<+\infty\) and a point \(x_1\in {\mathbb {S}}^2\) such that for any \(\varepsilon >0,\)

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _{B(x_{1},\varepsilon )}e^{u(t_k,\cdot )}}{\int _{{\mathbb {S}}^2}e^{u(t_k,\cdot )}}\ge 1-\varepsilon , \end{aligned}$$
(1.5)

i.e., the measure \(e^{u(t_k)}\) (after normalization) concentrates around one point on \({\mathbb {S}}^2\) (i.e., it resembles a one bubble). On the other hand, for the general supercritical case \(\rho >8\pi\) the blow-up analysis is not carried out and we are missing the blow-up criteria. One of our aims is to substantially refine the latter analysis and to give general blow-up criteria, see Corollary 1.5. As a matter of fact, this will follow by the analysis we will develop for more general problems: the sinh-Gordon equation (1.1) and Toda systems (1.7).

Let us return now to the sinh-Gordon equation (1.2) and its associated wave equation (1.1). In the last decades, the analysis concerning (1.3) was generalized for treating the sinh-Gordon equation (1.2) and we refer to [2, 3, 23, 24, 28, 39] for blow-up analysis, to [17] for uniqueness aspects and to [6, 18,19,20] for what concerns existence results. On the other hand, for what concerns the wave equation associated with (1.2), i.e., (1.1), there are few results mainly focusing on traveling wave solutions, see, for example, [1, 16, 37, 47, 51]. One of our aims is to develop the analysis for (1.1) in the spirit of [11] and to refine it with some new arguments. More precisely, by exploiting the variational analysis derived for Eq. (1.2), see in particular [6], we will prove global existence in time for (1.1) for the subcritical case and we will give general blow-up criteria for the supercritical and critical case. The sub/supercritical case refers to the sharp constant of the associated Moser–Trudinger inequality, as it will be clear in the sequel.

Before stating the results, let us fix some notation. Given \(T>0\) and a metric space X, we will denote C([0, T]; X) by \(C_T(X)\). \(C^k_T(X)\) and \(L^k_T(X)\), \(k\ge 1\), are defined in an analogous way. When we are taking time derivative for \(t\in [0,T]\), we are implicitly taking right (resp. left) derivative at the endpoint \(t=0\) (resp. \(t=T\)). When it will be clear from the context, we will simply write \(H^1, L^2\) to denote \(H^1(M), L^2(M)\), respectively, and

$$\begin{aligned} \Vert u\Vert _{H^1(M)}^2=\Vert \nabla u\Vert _{L^2(M)}^2+\Vert u\Vert _{L^2(M)}^2. \end{aligned}$$

Our first main result is to show that the initial value problem for (1.1) is locally well-posed in \(H^1\times L^2\).

Theorem 1.1

Let \(\rho _1,\rho _2\in {\mathbb {R}}\). Then, for any \((u_0,u_1)\in H^1(M)\times L^2(M)\) such that \(\int _{M}u_1=0\), there exist \(T=T(\rho _1, \rho _2, \Vert u_0\Vert _{H^1}, \Vert u_1\Vert _{L^2})>0\) and a unique, stable solution, i.e., depending continuously on \((u_0,u_1)\),

$$\begin{aligned} u:[0,T]\times M\rightarrow {\mathbb {R}}, \quad u\in C_T(H^1)\cap C_T^1(L^2), \end{aligned}$$

of (1.1) with initial data

$$\begin{aligned} \left\{ \begin{array}{l} u(0,\cdot )=u_0, \\ \partial _t u(0,\cdot )=u_1. \end{array} \right. \end{aligned}$$

Furthermore,

$$\begin{aligned} \int _{M}u(t,\cdot )=\int _{M}u_0 \quad {\text{ for }}\,\, {\text{ all }}\,\, t\in [0,T]. \end{aligned}$$
(1.6)

Remark 1.2

The assumption on the initial datum \(u_1\) to have zero average guarantees that the average (1.6) of the solution \(u(t,\cdot )\) to (1.1) is preserved in time. A consequence of the latter property is that the energy \(E(u(t,\cdot ))\) given in (3.9) is preserved in time as well, which will be then crucially used in the sequel, see the discussion later on.

The proof is based on a fixed point argument and the standard Moser–Trudinger inequality (2.1), see Sect. 3. Once the local existence is established, we address the existence of a global solution to (1.1). Indeed, by exploiting an energy argument jointly with the Moser–Trudinger inequality associated with (1.2), see (2.2), we deduce our second main result.

Theorem 1.3

Suppose \(\rho _1,\rho _2<8\pi .\) Then, for any \((u_0,u_1)\in H^1(M)\times L^2(M)\) such that \(\int _{M}u_1=0\), there exists a unique global solution \(u\in C({\mathbb {R}}^+;H^1)\cap C^1({\mathbb {R}}^+;L^2)\) of (1.1) with initial data \((u_0,u_1)\).

The latter case \(\rho _1,\rho _2<8\pi\) is referred as the subcritical case in relation to the sharp constant \(8\pi\) in the Moser–Trudinger inequality (2.2). The critical and supercritical case in which \(\rho _i\ge 8\pi\) for some i is subtler since the solutions to (1.2) might blow up. However, by exploiting the recent analysis concerning (1.2), see in particular [6], based on improved versions of the Moser–Trudinger inequality, see Proposition 2.1, we are able to give quite general blow-up criteria for (1.1). Our third main result is the following.

Theorem 1.4

Suppose \(\rho _i\ge 8\pi\) for some i. Let \((u_0,u_1)\in H^1(M)\times L^2(M)\) be such that \(\int _{M}u_1=0\) and let u be the solution of (1.1) obtained in Theorem 1.1. Suppose that u exists in \([0,T_0)\) for some \(T_0<+\infty\) and it cannot be extended beyond \(T_0\). Then, there exists a sequence \(t_k\rightarrow T_0^{-}\) such that

$$\begin{aligned} \lim _{k\rightarrow +\infty }\Vert \nabla u(t_k,\cdot )\Vert _{L^2}=+\infty , \quad \lim _{k\rightarrow +\infty }\max \left\{ \int _{M}e^{u(t_k,\cdot )}, \int _{M}e^{-u(t_k,\cdot )}\right\} =+\infty . \end{aligned}$$

Furthermore, if \(\rho _1\in [8m_1\pi ,8(m_1+1)\pi )\) and \(\rho _2\in [8m_2\pi ,8(m_2+1)\pi )\) for some \(m_1,m_2\in {\mathbb {N}}\), then there exist points \(\{x_1,\dots ,x_{m_1}\}\subset M\) such that for any \(\varepsilon >0,\) either

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _{\bigcup _{l=1}^{m_1}B(x_{l},\varepsilon )}e^{u(t_k,\cdot )}}{\int _{M}e^{u(t_k,\cdot )}}\ge 1-\varepsilon , \end{aligned}$$

or there exist points \(\{y_1,\dots ,y_{m_2}\}\subset M\) such that for any \(\varepsilon >0,\)

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _{\bigcup _{l=1}^{m_2}B(y_{l},\varepsilon )}e^{-u(t_k,\cdot )}}{\int _{M}e^{-u(t_k,\cdot )}}\ge 1-\varepsilon . \end{aligned}$$

The latter result shows that once the two parameters \(\rho _1, \rho _2\) are fixed in a critical or supercritical regime, the finite time blowup of the solutions to (1.1) yields the following alternative: either the measure \(e^{u(t_k)}\) (after normalization) concentrates around (at most) \(m_1\) points on M (i.e., it resembles a \(m_1\)-bubble) or \(e^{-u(t_k)}\) concentrates around \(m_2\) points on M. We point out that this is new for the mean field equation (1.4) as well and generalizes the previous blow-up criterion (1.5) obtained in [11] for \(\rho =8\pi\). More precisely, the general blow-up criteria for the supercritical mean field equation are the following.

Corollary 1.5

Suppose \(\rho \in [8m\pi ,8(m+1)\pi )\) for some \(m\in {\mathbb {N}}\), \(m\ge 1\). Let \((u_0,u_1)\in H^1(M)\times L^2(M)\) be such that \(\int _{M}u_1=0\), and let u be a solution of (1.4), where \({\mathbb {S}}^2\) is replaced by a compact surface M. Suppose that u exists in \([0,T_0)\) for some \(T_0<+\infty\) and it cannot be extended beyond \(T_0\). Then, there exist a sequence \(t_k\rightarrow T_0^{-}\) and m points \(\{p_1,\dots ,p_m\}\subset M\) such that for any \(\varepsilon >0\),

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\int _{\bigcup _{l=1}^{m}B(p_{l},\varepsilon )}e^{u(t_k,\cdot )}}{\int _{M}e^{u(t_k,\cdot )}} \ge 1-\varepsilon . \end{aligned}$$

Finally, it is worth to point out some possible generalizations of the results so far.

Remark 1.6

We may consider the following more general weighted problem

$$\begin{aligned} \partial _t^2u-\Delta _gu=\rho _1\left( \frac{h_1e^u}{\int _{M}h_1e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{h_2e^{-u}}{\int _{M}h_2e^{-u}}-\frac{1}{|M|}\right) , \end{aligned}$$

where \(h_i=h_i(x)\) are two smooth functions such that \(\frac{1}{C} \le h_i \le C\) on M, \(i=1,2\), for some \(C>0\). It is easy to check that Theorems 1.1, 1.3 and 1.4 extend to this case as well. The same argument applies also to the Toda system (1.7).

On the other hand, motivated by several applications in mathematical physics [1, 40, 42] we may consider the following asymmetric sinh-Gordon wave equation

$$\begin{aligned} \partial _t^2u-\Delta _gu=\rho _1\left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-au}}{\int _{M}e^{-au}}-\frac{1}{|M|}\right) , \end{aligned}$$

with \(a>0\). For \(a=2\), which corresponds to the Tzitzéica equation, we can exploit the detailed analysis in [25] to derive Theorems 1.1, 1.3 and 1.4 for this case as well (with suitable modifications accordingly to the associated Moser–Trudinger inequality). On the other hand, for general \(a>0\) the complete analysis is still missing and we can rely, for example, on [21] to get at least the existence results of Theorems 1.1 and 1.3.

We next consider the wave equation associated with some general Toda system,

$$\begin{aligned} \partial _t^2u_i-\Delta _gu_i=\sum _{j=1}^na_{ij}\,\rho _j\left( \frac{e^{u_j}}{\int _{M}e^{u_j}}-\frac{1}{|M|}\right) \quad {\text{ on }}\,\, M, \quad i=1,\dots ,n, \end{aligned}$$
(1.7)

where \(\rho _i\), \(i=1,\dots ,n\) are real parameters and \(A_n=(a_{ij})_{n\times n}\) is the following rank n Cartan matrix for \(SU(n+1)\):

$$\begin{aligned} {A}_n=\left( \begin{matrix} 2 &\quad -1 &\quad 0 &\quad \cdots &\quad 0 \\ -1 &\quad 2 &\quad -1 &\quad \cdots &\quad 0 \\ \vdots &\quad \vdots &\quad \vdots & \quad \ddots &\quad \vdots \\ 0 &\quad \cdots &\quad -1 & \quad 2 & \quad -1 \\ 0 &\quad \cdots & \quad 0 &\quad -1 &\quad 2 \end{matrix}\right) . \end{aligned}$$
(1.8)

The stationary equation related to (1.7) is the following Toda system:

$$\begin{aligned} -\Delta _gu_i=\sum _{j=1}^na_{ij}\,\rho _j\left( \frac{e^{u_j}}{\int _{M}e^{u_j}}-\frac{1}{|M|}\right) , \quad i=1,\dots ,n, \end{aligned}$$
(1.9)

which has been extensively studied since it has several applications both in mathematical physics and in geometry, for example, non-abelian Chern–Simons theory [15, 46, 50] and holomorphic curves in \({\mathbb {C}}{\mathbb {P}}^n\) [7, 10, 32]. There are by now many results concerning Toda-type systems in particular regarding existence of solutions [6, 22, 34], blow-up analysis [26, 31] and classification issues [32].

On the other hand, only partial results concerning the wave equation associated with the Toda system (1.7) were obtained in [11] which we recall here. First, the local well-posedness of (1.7) analogously as in Theorem 1.1 is derived for a general \(n\times n\) symmetric matrix \(A_n\). Second, by assuming \(A_n\) to be a positive definite symmetric matrix with nonnegative entries the authors were able to deduce a global existence result by exploiting a Moser–Trudinger-type inequality suitable for this setting, see [44]. On the other side, no results are available neither for mixed positive and negative entries of the matrix \(A_n\) (which are relevant in mathematical physics and in geometry, see, for example, the above Toda system) nor for blow-up criteria. Our aim is to complete the latter analysis.

Before stating the results, let us fix some notation for the system setting. We denote the product space as \((H^1(M))^n=H^1(M)\times \dots \times H^1(M)\). To simplify the notation, to take into account an element \((u_1,\dots ,u_n)\in (H^1(M))^n\) we will rather write \(H^1(M)\ni {\mathbf {u}}: M\mapsto (u_1,\dots ,u_n)\in {\mathbb {R}}^n\). With a little abuse of notation, we will write \(\int _{M} {\mathbf {u}}\) when we want to consider the integral of each component \(u_i\), \(i=1,\dots ,n\).

Since the local well-posedness of (1.7) is already known from [11], our first result concerns the global existence in time.

Theorem 1.7

Suppose \(\rho _i<4\pi\) for all \(i=1,\dots ,n.\) Then, for any \(({\mathbf {u}}_0,{\mathbf {u}}_1)\in H^1(M)\times L^2(M)\) such that \(\int _{M}{\mathbf {u}}_1=0\), there exists a unique global solution

$$\begin{aligned} {\mathbf {u}}:{\mathbb {R}}^+\times M\rightarrow {\mathbb {R}}^n, \quad {\mathbf {u}}\in C({\mathbb {R}}^+;H^1)\cap C^1({\mathbb {R}}^+;L^2), \end{aligned}$$

of (1.7) with initial data

$$\begin{aligned} \left\{ \begin{array}{l} {\mathbf {u}}(0,\cdot )={\mathbf {u}}_0, \\ \partial _t {\mathbf {u}}(0,\cdot )={\mathbf {u}}_1. \end{array} \right. \end{aligned}$$

The latter result follows by an energy argument and a Moser–Trudinger-type inequality for systems as obtained in [27]. On the other hand, when \(\rho _i\ge 4\pi\) for some i, the Moser–Trudinger inequality does not give any control and the solutions of (1.9) might blow up. In the latter case, by exploiting improved versions of the Moser–Trudinger inequality for the system recently derived in [5] we are able to give the following general blow-up criteria.

Theorem 1.8

Suppose \(\rho _i\ge 4\pi\) for some i. Let \(({\mathbf {u}}_0,{\mathbf {u}}_1)\in H^1(M)\times L^2(M)\) be such that \(\int _{M}{\mathbf {u}}_1=0\), and let \({\mathbf {u}}\) be the solution of (1.7). Suppose that \({\mathbf {u}}\) exists in \([0,T_0)\) for some \(T_0<\infty\) and it cannot be extended beyond \(T_0\). Then, there exists a sequence \(t_k\rightarrow T_0^{-}\) such that

$$\begin{aligned} \lim _{k\rightarrow +\infty } \max _j \Vert \nabla u_j(t_k,\cdot )\Vert _{L^2}=+\infty , \quad \lim _{k\rightarrow +\infty }\max _j\int _{M}e^{u_j(t_k,\cdot )}=+\infty . \end{aligned}$$

Furthermore, if \(\rho _i\in [4m_i\pi ,4(m_i+1)\pi )\) for some \(m_i\in {\mathbb {N}}\), \(i=1,\dots ,n\), then there exist at least one index \(j\in \{1,\dots ,n\}\) and \(m_j\) points \(\{x_{j,1},\dots ,x_{j,m_j}\}\in M\) such that for any \(\varepsilon >0\),

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{\int _{\bigcup _{l=1}^{m_j}B(x_{j,l},\varepsilon )}e^{u_j(t_k,\cdot )}}{\int _{M}e^{u_j(t_k,\cdot )}} \ge 1-\varepsilon . \end{aligned}$$

Therefore, for the finite time blow-up solutions to (1.7) there exists at least one component \(u_j\) such that the measure \(e^{u(t_k)}\) (after normalization) concentrates around (at most) \(m_j\) points on M. One can compare this result with the one for the sinh-Gordon equation (1.1) or the mean field equation (1.4), see Theorem 1.4 and Corollary 1.5, respectively.

Finally, we have the following possible generalization of the system (1.7).

Remark 1.9

We point out that since the improved versions of the Moser–Trudinger inequality in [5] hold for general symmetric, positive definite matrices \(A_n=(a_{ij})_{n\times n}\) with non-positive entries outside the diagonal, we can derive similar existence results and blow-up criteria as in Theorems 1.7, 1.8, respectively, for this general class of matrices as well. In particular, after some simple transformations (see, for example, the introduction in [2]) we may treat the following Cartan matrices:

$$\begin{aligned} B_n&= \left( \begin{matrix} 2 &\quad -1 & \quad 0 &\quad \cdots &\quad 0 \\ -1 &\quad 2 &\quad -1 &\quad \cdots &\quad 0 \\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\ 0 &\quad \cdots &\quad -1 &\quad 2 &\quad -2 \\ 0 &\quad \cdots &\quad 0 &\quad -1 &\quad 2 \end{matrix}\right) , \quad {C}_n=\left( \begin{matrix} 2 &\quad -1 &\quad 0 &\quad \cdots &\quad 0 \\ -1 &\quad 2 &\quad -1 & \quad \cdots &\quad 0 \\ \vdots &\quad \vdots &\quad \vdots &\quad \ddots &\quad \vdots \\ 0 & \quad \cdots &\quad -1 &\quad 2 &\quad -1 \\ 0 &\quad \cdots &\quad 0 &\quad -2 & \quad 2 \end{matrix}\right) , \\ {G}_2&= \left( \begin{matrix} 2&\quad -1\\ -3&\quad 2 \end{matrix}\right) , \end{aligned}$$

which are relevant in mathematical physics, see, for example, [15]. To simplify the presentation, we give the details just for the matrix \(A_n\) in (1.8).

The paper is organized as follows. In Sect. 2, we collect some useful results, and in Sect. 3, we prove the main results of this paper: local well-posedness, global existence and blow-up criteria.

2 Preliminaries

In this section, we collect some useful results concerning the stationary sinh-Gordon equation (1.2), Toda system (1.9) and the solutions of wave equations which will be used in the proof of the main results in the next section.

In the sequel, the symbol \(\overline{u}\) will denote the average of u, that is

$$\begin{aligned} \overline{u}= \fint _{M} u=\frac{1}{|M|}\int _{M} u. \end{aligned}$$

Let us start by recalling the well-known Moser–Trudinger inequality

$$\begin{aligned} 8\pi \log \int _{M} e^{u-\overline{u}} \le \frac{1}{2} \int _{M} |\nabla u|^2 + C_{(M,g)}\,, \quad u\in H^1(M). \end{aligned}$$
(2.1)

For the sinh-Gordon equation (1.2), a similar sharp inequality was obtained in [39],

$$\begin{aligned} 8\pi \left( \log \int _{M} e^{u-\overline{u}} + \log \int _{M} e^{-u+\overline{u}}\right) \le \frac{1}{2} \int _{M} |\nabla u|^2 + C_{(M,g)}\,,\quad u\in H^1(M). \end{aligned}$$
(2.2)

We recall now some of main features concerning the variational analysis of the sinh-Gordon equation (1.2) recently derived in [6], which will be exploited later on. First of all, letting \(\rho _1, \rho _2\in {\mathbb {R}}\) the associated Euler–Lagrange functional for Eq. (1.2) is given by \(J_{\rho _1,\rho _2}:H^1(M)\rightarrow {\mathbb {R}}\),

$$\begin{aligned} J_{\rho _1,\rho _2}(u)=\frac{1}{2}\int _{M}|\nabla u|^2-\rho _1\log \int _{M}e^{u-\overline{u}} -\rho _2\log \int _{M}e^{-u+\overline{u}}. \end{aligned}$$
(2.3)

Observe that if \(\rho _1, \rho _2\le 8\pi\), by (2.2), we readily have

$$\begin{aligned} J_{\rho _1,\rho _2}(u)\ge -C, \end{aligned}$$

for any \(u\in H^1(M)\), where \(C>0\) is a constant independent of u. On the other hand, as soon as \(\rho _i>8\pi\) for some \(i=1,2\) the functional \(J_{\rho _1,\rho _2}\) is unbounded from below. To treat the latter supercritical case, one needs improved versions of the Moser–Trudinger inequality (2.2) which roughly assert that the more the measures \(e^u, e^{-u}\) are spread over the surface, the bigger is the constant in the left-hand side of (2.2). More precisely, we have the following result.

Proposition 2.1

([6]) Let \(\delta , \theta >0\), \(k,l\in {\mathbb {N}}\) and \(\{\Omega _{1,i},\Omega _{2,j}\}_{i\in \{1,\dots ,k\},j\in \{1,\dots ,l\}}\subset M\) be such that

$$\begin{aligned}&d(\Omega _{1,i},\Omega _{1,i'})\ge \delta ,\quad \forall \, i,i'\in \{1,\dots ,k\}, \, i\ne i', \\&d(\Omega _{2,j},\Omega _{2,j'})\ge \delta ,\quad \forall \, j,j' \in \{1,\dots ,l\}, \, j\ne j'. \end{aligned}$$

Then, for any \(\varepsilon >0\) there exists \(C=C\left( \varepsilon ,\delta ,\theta ,k,l,M\right)\) such that if \(u\in H^1(M)\) satisfies

$$\begin{aligned} \int _{\Omega _{1,i}} e^{u} \ge \theta \int _{M} e^{u}, \,\, \forall i\in \{1,\dots ,k\}, \qquad \int _{\Omega _{2,j}} e^{-u} \ge \theta \int _{M} e^{-u}, \,\, \forall j\in \{1,\dots ,l\}, \end{aligned}$$

it follows that

$$\begin{aligned} 8k\pi \log \int _{M} e^{u-\overline{u}}+8l\pi \log \int _{M} e^{-u+\overline{u}}\le \frac{1+\varepsilon }{2}\int _{M} |\nabla u|^2\,{\mathrm{d}}V_g+C. \end{aligned}$$

From the latter result, one can deduce that if the \(J_{\rho _1,\rho _2}(u)\) is large negative at least one of the two measures \(e^u, e^{-u}\) has to concentrate around some points of the surface.

Proposition 2.2

([6]) Suppose \(\rho _i\in (8m_i\pi ,8(m_i+1)\pi )\) for some \(m_i\in {\mathbb {N}}\), \(i=1,2\) (\(m_i\ge 1\) for some \(i=1,2\)). Then, for any \(\varepsilon , r>0\) there exists \(L=L(\varepsilon ,r)\gg 1\) such that for any \(u\in H^1(M)\) with \(J_{\rho _1,\rho _2}(u)\le -L\), there are either some \(m_1\) points \(\{x_1,\dots ,x_{m_1}\}\subset M\) such that

$$\begin{aligned} \frac{\int _{\cup _{l=1}^{m_1}B_r(x_l)}e^{u}}{\int _{M}e^u}\ge 1-\varepsilon , \end{aligned}$$

or some \(m_2\) points \(\{y_1,\dots ,y_{m_2}\}\subset M\) such that

$$\begin{aligned} \frac{\int _{\cup _{l=1}^{m_2}B_r(y_l)}e^{-u}}{\int _{M}e^{-u}}\ge 1-\varepsilon . \end{aligned}$$

We next briefly recall some variational aspects of the stationary Toda system (1.9). Recall the matrix \(A_n\) in (1.8) and the notation of \({\mathbf {u}}\) introduced before Theorem 1.7 and write \(\rho =(\rho _1,\dots ,\rho _n)\). The associated functional for the system (1.9) is given by \(J_{\rho }:H^1(M)\rightarrow {\mathbb {R}}\),

$$\begin{aligned} J_{\rho }(\mathbf {u})=\frac{1}{2}\int _{M}\sum _{i,j=1}^n a^{ij}\langle \nabla u_i,\nabla u_j\rangle -\sum _{i=1}^n\rho _i\log \int _{M}e^{u_i-\overline{u}_i}, \end{aligned}$$
(2.4)

where \((a^{ij})_{n\times n}\) is the inverse matrix \(A_n^{-1}\) of \(A_n\). A Moser–Trudinger inequality for (2.4) was obtained in [27], which asserts that

$$\begin{aligned} J_{\rho }({\mathbf {u}})\ge C, \end{aligned}$$
(2.5)

for any \({\mathbf {u}}\in H^1(M)\), where C is a constant independent of \({\mathbf {u}}\), if and only if \(\rho _i\le 4\pi\) for any \(i=1,\dots ,n.\) In particular, if \(\rho _i>4\pi\) for some \(i=1,\dots ,n\) the functional \(J_\rho\) is unbounded from below. As for the sinh-Gordon equation (1.2), we have improved versions of the Moser–Trudinger inequality (2.5) recently derived in [5] (see also [6]) which yield concentration of the measures \(e^{u_j}\) whenever \(J_{\rho }({\mathbf {u}})\) is large negative.

Proposition 2.3

([5, 6]) Suppose \(\rho _i\in (4m_i\pi ,4(m_i+1)\pi )\) for some \(m_i\in {\mathbb {N}},~i=1,\dots ,n\) (\(m_i\ge 1\) for some \(i=1,\dots ,n\)). Then, for any \(\varepsilon , r>0\) there exists \(L=L(\varepsilon ,r)\gg 1\) such that for any \({\mathbf {u}}\in H^1(M)\) with \(J_{\rho }({\mathbf {u}})\le -L\), there exists at least one index \(j\in \{1,\dots ,n\}\) and \(m_j\) points \(\{x_1,\dots ,x_{m_j}\}\subset M\) such that

$$\begin{aligned} \frac{\int _{\cup _{l=1}^{m_j}B_r(x_l)}e^{u_j}}{\int _{M}e^{u_j}}\ge 1-\varepsilon . \end{aligned}$$

Finally, let us state a standard result concerning the wave equation, that is the Duhamel principle. Let us first recall that every function in \(L^2(M)\) can be decomposed as a convergent sum of eigenfunctions of the Laplacian \(\Delta _g\) on M. Then, one can define the operators \(\cos (\sqrt{-\Delta _g})\) and \(\frac{\sin (\sqrt{-\Delta _g})}{\sqrt{-\Delta _g}}\) acting on \(L^2(M)\) using the spectral theory. Consider now the initial value problem

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t^2v-\Delta _gv=f(t,x),\\ v(0,\cdot )=u_0,\quad \partial _tv(0,\cdot )=u_1, \end{array}\right. \end{aligned}$$
(2.6)

on \([0,+\infty )\times M\). Recall the notation of \(C_T(X)\) and \({\mathbf {u}}\) before Theorems 1.1 and 1.7, respectively. Then, the following Duhamel formula holds true.

Proposition 2.4

Let \(T>0\), \((u_0,u_1)\in H^1(M)\times L^2(M)\) and let \(f\in L^1_T(L^2(M))\). Then, (2.6) has a unique solution

$$\begin{aligned} v:[0,T)\times M\rightarrow {\mathbb {R}}, \quad v\in C_T(H^1)\cap C_T^1(L^2), \end{aligned}$$

given by

$$\begin{aligned} v(t,x)&=\cos \left( t\sqrt{-\Delta _g}\right) u_0+\frac{\sin (t\sqrt{-\Delta _g})}{\sqrt{-\Delta _g}}\,u_1\nonumber \\&\quad +\int _0^t\frac{\sin \bigr ((t-s)\sqrt{-\Delta _g}\bigr )}{\sqrt{-\Delta _g}}\,f(s)\,{\mathrm {d}}s. \end{aligned}$$
(2.7)

Furthermore, it holds

$$\begin{aligned} \Vert v\Vert _{C_T(H^1)}+\Vert \partial _tv\Vert _{C_T(L^2)}\le 2\left( \Vert u_0\Vert _{H^1}+\Vert u_1\Vert _{L^2}+\Vert f\Vert _{L_T^1(L^2)}\right) . \end{aligned}$$
(2.8)

The same results hold as well if \(u_0,u_1,\) and \(f(t,\cdot )\) are replaced by \({\mathbf {u}}_0, {\mathbf {u}}_1\) and \({\mathbf {f}}(t,\cdot )\), respectively.

3 Proof of the main results

In this section, we derive the main results of the paper, that is local well-posedness, global existence and blow-up criteria for the wave sinh-Gordon equation (1.1), see Theorems 1.1, 1.3 and 1.4, respectively. Since the proofs of global existence and blow-up criteria for the wave Toda system (1.7) (Theorems 1.7, 1.8) are obtained by similar arguments, we will present full details for what concerns the wave sinh-Gordon equation and point out the differences in the two arguments, where necessary.

3.1 Local and global existence

We start by proving the local well-posedness of the wave sinh-Gordon equation (1.1). The proof is mainly based on a fixed point argument and the Moser–Trudinger inequality (2.1).

Proof of Theorem 1.1

Let \((u_0,u_1)\in H^1(M)\times L^2(M)\) be such that \(\int _{M}u_1=0\). Take \(T>0\) to be fixed later on. We set

$$\begin{aligned} R=3\left( \Vert u_0\Vert _{H^1}+\Vert u_1\Vert _{L^2}\right) ,\quad I=\fint _{ M}u_0=\frac{1}{|M|}\int _{M}u_0, \end{aligned}$$
(3.1)

and we introduce the space \(B_T\) given by

$$\begin{aligned} B_T=\left\{ u\in C_T(H^1(M))\cap C_T^1(L^2(M)) \,:\, \Vert u\Vert _*\le R,~ \fint _{M}u(s,\cdot )=I \, {\text{ for }}\,\, {\text{ all }}\,\, s\in [0,T]\right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert u\Vert _*=\Vert u\Vert _{C_T(H^1)}+\Vert \partial _tu\Vert _{C_T(L^2)}. \end{aligned}$$

For \(u\in B_T\), we consider the initial value problem

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t^2v-\Delta _gv=f(s,x)=\rho _1\left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-u}}{\int _{M}e^{-u}}-\frac{1}{|M|}\right) , \\ v(0,\cdot )=u_0,\quad \partial _tv(0,\cdot )=u_1, \end{array}\right. \end{aligned}$$
(3.2)

on \([0,T]\times M\). Applying Proposition 2.4, we deduce the existence of a unique solution of (3.2).

Step 1 We aim to show that \(v\in B_{T}\) if T is taken sufficiently small. Indeed, still by Proposition 2.4 we have

$$\begin{aligned} \begin{aligned} \Vert v\Vert _*&\le 2\left( \Vert u_0\Vert _{H^1}+\Vert u_1\Vert _{L^2}\right) +2\rho _1\int _0^T\left\| \left( \frac{e^u}{\int _{M}e^u}-\frac{1}{|M|}\right) \right\| _{L^2}{\mathrm {d}}s\\&\quad +2\rho _2\int _0^T\left\| \left( \frac{e^{-u}}{\int _{M}e^{-u}}-\frac{1}{|M|}\right) \right\| _{L^2}{\mathrm {d}}s. \end{aligned} \end{aligned}$$
(3.3)

Since \(u\in B_T\), we have \(\fint _{M}u(s,\cdot )=I\) for all \(s\in [0,T]\), and therefore, by the Jensen inequality,

$$\begin{aligned} \fint _{M}e^{u}\ge e^{\fint _{M}u}=e^{I}\quad {\text{ and }}\,\,\quad \fint _{M}e^{-u}\ge e^{-\fint _{M}u}=e^{-I}. \end{aligned}$$

Therefore, we can bound the last two terms on the right-hand side of (3.3) by

$$\begin{aligned} CT(|\rho _1|+|\rho _2|)+CT|\rho _1|e^{-I}\max _{s\in [0,T]}\Vert e^{u(s,\cdot )}\Vert _{L^2}+CT|\rho _2|e^{I}\max _{s\in [0,T]}\Vert e^{-u(s,\cdot )}\Vert _{L^2}, \end{aligned}$$

for some \(C>0\). On the other hand, recalling the Moser–Trudinger inequality (2.1), we have for \(s\in [0,T]\)

$$\begin{aligned} \begin{aligned} \Vert e^{u(s,\cdot )}\Vert _{L^2}^2&=\int _{M}e^{2u(s,\cdot )}=\int _{M}e^{2(u(s,\cdot )-\overline{u})}e^{2I} \\ &\le C\exp \left( \frac{1}{4\pi }\int _{M}|\nabla u(s,\cdot )|^2\right) e^{2I} \le Ce^{2I}e^{\frac{1}{4\pi }R^2}, \end{aligned} \end{aligned}$$
(3.4)

for some \(C>0\), where we used \(\Vert u\Vert _*\le R\). Similarly, we have

$$\begin{aligned} \Vert e^{-u(s,\cdot )}\Vert _{L^2}^2\le Ce^{-2I}e^{\frac{1}{4\pi }R^2}. \end{aligned}$$

Hence, recalling the definition of R in (3.1), by (3.3) and the above estimates we conclude

$$\begin{aligned} \Vert v\Vert _*&\le 2\left( \Vert u_0\Vert _{H^1}+\Vert u_1\Vert _{L^2}\right) +CT(|\rho _1|+|\rho _2|)+CT(|\rho _1|+|\rho _2|)e^{\frac{1}{8\pi }R^2} \\&= \frac{2}{3} R+CT(|\rho _1|+|\rho _2|)+CT(|\rho _1|+|\rho _2|)e^{\frac{1}{8\pi }R^2}. \end{aligned}$$

Therefore, if \(T>0\) is taken sufficiently small, \(T=T(\rho _1,\rho _2,\Vert u_0\Vert _{H^1},\Vert u_1\Vert _{L^2})\), then \(\Vert v\Vert _*\le R\).

Moreover, observe that if we integrate both sides of (3.2) on M, we get

$$\begin{aligned} \partial _t^2\overline{v}(s)=0, \quad {\text{ for }}\,\, {\text{ all }}\,\, s\in [0,T], \end{aligned}$$

and hence,

$$\begin{aligned} \partial _t\overline{v}(s)=\partial _t\overline{v}(0)=\overline{u}_1=0, \quad {\text{ for }}\,\, {\text{ all }}\,\, s\in [0,T]. \end{aligned}$$

It follows that

$$\begin{aligned} \fint _{M}v(s,\cdot )=\fint _{M}v(0,\cdot )=\fint _{M}u_0=I \quad {\text{ for }}\,\, {\text{ all }}\,\, s\in [0,T]. \end{aligned}$$

Thus, for this choice of T we conclude that \(v\in B_T\).

Therefore, we can define a map

$$\begin{aligned} {\mathcal {F}}:B_T\rightarrow B_T, \quad v={\mathcal {F}}(u). \end{aligned}$$

Step 2 We next prove that by taking a smaller T if necessary, \({\mathcal {F}}\) is a contraction. Indeed, let \(u_1,u_2\in B_T\) be such that \(v_i={\mathcal {F}}(u_i), \,i=1,2\). Then, \(v=v_1-v_2\) satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2v-\Delta _gv=\rho _1\left( \frac{e^{u_1}}{\int _{M}e^{u_1}}-\frac{e^{u_2}}{\int _{M}e^{u_2}}\right) -\rho _2\left( \frac{e^{-u_1}}{\int _{M}e^{-u_1}}-\frac{e^{-u_2}}{\int _{M}e^{-u_2}}\right) ,\\ v(0,\cdot )=0,\quad \partial _tv(0,\cdot )=0. \end{array}\right. } \end{aligned}$$

Hence, by Proposition 2.4 we have

$$\begin{aligned} \begin{aligned} \Vert v\Vert _*&\le 2\rho _1\int _0^T\left\| \left( \frac{e^{u_1}}{\int _{ M}e^{u_1}} -\frac{e^{u_2}}{\int _{ M}e^{u_2}}\right) \right\| _{L^2}{\mathrm {d}}s\\&\quad +2\rho _2\int _0^T\left\| \left( \frac{e^{-u_1}}{\int _{ M}e^{-u_1}} -\frac{e^{-u_2}}{\int _{ M}e^{-u_2}}\right) \right\| _{L^2}{\mathrm {d}}s. \end{aligned} \end{aligned}$$
(3.5)

For \(s\in [0,T]\), we use the following decomposition,

$$\begin{aligned} \left\| \left( \frac{e^{u_1(s,\cdot )}}{\int _{ M}e^{u_1(s,\cdot )}} -\frac{e^{u_2(s,\cdot )}}{\int _{ M}e^{u_2(s,\cdot )}}\right) \right\| _{L^2}&\le \left\| \frac{e^{u_1}-e^{u_2}}{\int _{ M}e^{u_1}}\right\| _{L^2}\nonumber \\&\quad+\left\| \frac{e^{u_2}\left( \int _{ M}e^{u_1}-\int _{ M}e^{u_2}\right) }{\left( \int _{ M}e^{u_1}\right) \left( \int _{ M}e^{u_2}\right) }\right\| _{L^2}. \end{aligned}$$
(3.6)

Reasoning as before, the first term in the right-hand side of the latter estimate is bounded by

$$\begin{aligned}&Ce^{-I}\left\| (u_1(s,\cdot )-u_2(s,\cdot ))(e^{_1(s,\cdot )}+e^{u_2(s,\cdot )})\right\| _{L^2} \nonumber \\&\quad \le C e^{-I}\Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{L^4}\left( \Vert e^{u_1(s,\cdot )}\Vert _{L^4}+\Vert e^{u_2(s,\cdot )}\Vert _{L^4}\right) , \end{aligned}$$
(3.7)

for some \(C>0\), where we used the Hölder inequality. Moreover, we have

$$\begin{aligned} \begin{aligned} \Vert e^{u_i(s,\cdot )}\Vert _{L^4}^4&=\int _{ M}e^{4(u_i(s,\cdot )-\overline{u}_1(s))}e^{4I} \le Ce^{4I}\exp \left( \frac{1}{\pi }\int _{ M}|\nabla u_i(s,\cdot )|^2\right) \\ &\le Ce^{4I}e^{\frac{1}{\pi }R^2}, \quad i=1,2, \end{aligned} \end{aligned}$$

for some \(C>0\). Using the latter estimate for the second term in (3.7) and the Sobolev inequality for the first term, we can bound (3.7) by

$$\begin{aligned} Ce^{\frac{1}{4\pi }R^2}\Vert u_1-u_2\Vert _{H^1} \end{aligned}$$

and hence

$$\begin{aligned} \left\| \frac{e^{u_1}-e^{u_2}}{\int _{ M}e^{u_1}}\right\| _{L^2} \le Ce^{\frac{1}{4\pi }R^2}\Vert u_1-u_2\Vert _{H^1}. \end{aligned}$$
(3.8)

On the other hand, by using (3.4), the second term in (3.6) is bounded by

$$\begin{aligned} \begin{aligned}&Ce^{-2 I}\Vert e^{u_2(s,\cdot )}\Vert _{L^2}\int _{ M}|u_1(s,\cdot )-u_2(s,\cdot )|\left( e^{u_1(s,\cdot )}+e^{u_2(s,\cdot )}\right) \\&\quad \le Ce^{-2I}\Vert e^{u_2(s,\cdot )}\Vert _{L^2} \left( \Vert e^{u_1(s,\cdot )}\Vert _{L^2}+\Vert e^{u_2(s,\cdot )}\Vert _{L^2}\right) \Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{L^2}\\&\quad \le Ce^{\frac{1}{4\pi }R^2}\Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{H^1}, \end{aligned} \end{aligned}$$

for some \(C>0\), where in the last step, we used the Sobolev inequality.

In conclusion, we have

$$\begin{aligned} \left\| \left( \frac{e^{u_1(s,\cdot )}}{\int _{ M}e^{u_1(s,\cdot )}} -\frac{e^{u_2(s,\cdot )}}{\int _{ M}e^{u_2(s,\cdot )}}\right) \right\| _{L^2} \le Ce^{\frac{1}{4\pi }R^2}\Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{L^2}. \end{aligned}$$

Similarly,

$$\begin{aligned} \left\| \left( \frac{e^{-u_1(s,\cdot )}}{\int _{ M}e^{-u_1(s,\cdot )}} -\frac{e^{-u_2(s,\cdot )}}{\int _{ M}e^{-u_2(s,\cdot )}}\right) \right\| _{L^2} \le Ce^{\frac{1}{4\pi }R^2}\Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{L^2}. \end{aligned}$$

Finally, by the latter estimate, (3.8) and by (3.6), (3.5), we conclude that

$$\begin{aligned} \Vert v\Vert _*&\le CT(|\rho _1|+|\rho _2|)e^{\frac{1}{4\pi }R^2}\Vert u_1(s,\cdot )-u_2(s,\cdot )\Vert _{H^1} \\&\le CT(|\rho _1|+|\rho _2|)e^{\frac{1}{4\pi }R^2}\Vert u_1-u_2\Vert _*\,. \end{aligned}$$

Therefore, if \(T>0\) is taken sufficiently small, \(T=T(\rho _1,\rho _2,\Vert u_0\Vert _{H^1},\Vert u_1\Vert _{L^2})\), then \({\mathcal {F}}\) is a contraction map. The latter fact yields the existence of a unique fixed point for \({\mathcal {F}}\), which solves (1.1) with initial conditions \((u_0, u_1)\).

The same arguments with suitable adaptations show that the initial value problem (1.1) is locally well-posed, so we omit the details. The proof is completed. □

We next prove that if the two parameters in (1.1) are taken in a subcritical regime, then there exists a global solution to the initial value problem associated with (1.1). To this end, we will exploit an energy argument jointly with the Moser–Trudinger inequality related to (1.2), see (2.2). For a solution u(t, x) to (1.1), we define its energy as

$$\begin{aligned} E(u(t,\cdot ))=\frac{1}{2}\int _{ M}(|\partial _tu|^2+|\nabla u|^2)-\rho _1\log \int _{ M}e^{u-\overline{u}} -\rho _2\log \int _{ M}e^{-u+\overline{u}}, \end{aligned}$$
(3.9)

for \(t\in [0,T]\). We point out that

$$\begin{aligned} E(u(t,\cdot ))=\frac{1}{2}\int _{ M}|\partial _tu|^2 +J_{\rho _1,\rho _2}(u(t,\cdot )), \end{aligned}$$

where \(J_{\rho _1,\rho _2}\) is the functional introduced in (2.3). We first show that the latter energy is conserved in time along the solution u.

Lemma 3.1

Let \(\rho _1,\rho _2\in {\mathbb {R}}\) and let \((u_0,u_1)\in H^1( M)\times L^2( M)\) be such that \(\int _{ M}u_1=0\). Let \(u\in C_T(H^1)\cap C_T^1(L^2)\), for some \(T>0\), be a solution to (1.1) with initial data \((u_0, u_1)\) and let E(u) be defined in (3.9). Then, it holds

$$\begin{aligned} E(u(t,\cdot ))=E(u(0,\cdot )) \quad {\text{ for }}\,\, {\text{ all }}\,\, t\in [0,T]. \end{aligned}$$

Proof

We will show that

$$\begin{aligned} \partial _tE(u(t,\cdot ))=0 \quad {\text{ for }}\,\,{ \text{ all }}\,\, t\in [0,T]. \end{aligned}$$

We have

$$\begin{aligned} \begin{aligned} \partial _tE(u(t,\cdot ))=\int _{ M}(\partial _tu)\left( \partial _t^2u\right) +\int _{ M}\langle \nabla \partial _tu,\nabla u\rangle -\rho _1\frac{\int _{ M}e^{u}\partial _tu}{\int _{ M}e^{u}}+\rho _2\frac{\int _{ M}e^{-u}\partial _tu}{\int _{ M}e^{-u}}. \end{aligned} \end{aligned}$$
(3.10)

After integration by parts, the first two terms in the right-hand side of the latter equation give

$$\begin{aligned} \begin{aligned} \int _{ M}(\partial _tu)\left( \partial _t^2u-\Delta _gu\right) =\int _{ M}\partial _tu\left( \rho _1\left( \frac{e^u}{\int _{ M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-u}}{\int _{ M}e^{-u}}-\frac{1}{|M|}\right) \right) , \end{aligned} \end{aligned}$$

where we have used the fact that u satisfies (1.1). Plugging the latter equation into (3.10), we readily have

$$\begin{aligned} \partial _tE(u(t),\cdot )=\frac{\rho _2-\rho _1}{|M|}\int _{ M}\partial _tu=\frac{\rho _2-\rho _1}{|M|}\,\partial _t\left( \int _{ M}u\right) =0 \quad {\text{ for }} \,\,{\text{ all }}\,\, t\in [0,T], \end{aligned}$$

since \(\int _{ M}u(t,\cdot )=\int _{ M}u_0\) for all \(t\in [0,T]\), see Theorem 1.1. This concludes the proof. □

We can now prove the global existence result for (1.1) in the subcritical regime \(\rho _1,\rho _2<8\pi\).

Proof of Theorem 1.3

Suppose \(\rho _1, \rho _2<8\pi\). Let \((u_0,u_1)\in H^1( M)\times L^2( M)\) be such that \(\int _{ M}u_1=0\) and let u be the solution to (1.1) with initial data \((u_0,u_1)\) obtained in Theorem 1.1. Suppose that u exists in \([0,T_0)\). With a little abuse of notation, \(C([0,T_0);H^1)\) will be denoted here still by \(C_{T_0}(H^1)\). Analogously, we will use the notation \(C_{T_0}^1(L^2)\). We have that \(u\in C_{T_0}(H^1)\cap C_{T_0}^1(L^2)\) satisfy

$$\begin{aligned} \partial _t^2u-\Delta u=\rho _1\left( \frac{e^u}{\int _{ M}e^u}-\frac{1}{|M|}\right) -\rho _2\left( \frac{e^{-u}}{\int _{ M}e^{-u}}-\frac{1}{|M|}\right) \ \ {\text{ on }}\,\,[0,T_0)\times M. \end{aligned}$$

We claim that

$$\begin{aligned} \Vert u\Vert _{C_{T_0}(H^1)}+\Vert \partial _t u\Vert _{C_{T_0}(L^2)}\le C, \end{aligned}$$
(3.11)

for some \(C>0\) depending only on \(\rho _1,\rho _2\) and \((u_0,u_1)\). Once the claim is proven, we can extend the solution u for a fixed amount of time starting at any \(t\in [0,T_0)\), which in particular implies that the solution u can be extended beyond time \(T_0\). Repeating the argument, we can extend u for any time and obtain a global solution as desired.

Now, we shall prove (3.11). We start by recalling that the energy \(E(u(t,\cdot ))\) in (3.9) is conserved in time, see Lemma 3.1, that is,

$$\begin{aligned} E(u(t,\cdot ))=E(u(0,\cdot )) \quad {\text{ for }}\,\, {\text{ all }}\,\, t\in [0,T_0). \end{aligned}$$
(3.12)

Suppose first \(\rho _1,\rho _2\in (0,8\pi )\). By the Moser–Trudinger inequality (2.2), we have

$$\begin{aligned} 8\pi \left( \log \int _{ M}e^{u(t,\cdot )-\overline{u}(t)}+\log \int _{ M}e^{-u(t,\cdot )+\overline{u}(t)}\right) \le \frac{1}{2}\int _{ M}|\nabla u(t,\cdot )|^2+C, \quad t\in [0,T_0), \end{aligned}$$

where \(C>0\) is independent of \(u(t,\cdot )\). Observe moreover that by the Jensen inequality, it holds

$$\begin{aligned} \log \int _{ M}e^{u(t,\cdot )-\overline{u}(t)}\ge 0, \quad \log \int _{ M}e^{-u(t,\cdot )+\overline{u}(t)}\ge 0 , \quad t\in [0,T_0). \end{aligned}$$
(3.13)

Therefore, letting \(\rho =\max \{\rho _1,\rho _2\}\) we have

$$\begin{aligned} E(u(t,\cdot ))&\ge \frac{1}{2}\int _{ M}(|\partial _tu(t,\cdot )|^2+|\nabla u(t,\cdot )|^2) \nonumber \\&\quad -\rho \left( \log \int _{ M}e^{u(t,\cdot )-\overline{u}(t)} -\log \int _{ M}e^{-u(t,\cdot )+\overline{u}(t)}\right) \nonumber \\&\ge \frac{1}{2}\int _{ M}(|\partial _tu(t,\cdot )|^2+|\nabla u(t,\cdot )|^2)-\frac{\rho }{16\pi }\int _{ M} |\nabla u(t,\cdot )|^2 - C\rho , \end{aligned}$$
(3.14)

for \(t\in [0,T_0)\), where \(C>0\) is independent of \(u(t,\cdot )\). Finally, since \(\rho <8\pi\) and by using (3.12), we deduce

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left( 1-\frac{\rho }{8\pi }\right) \left( \Vert \partial _tu(t,\cdot )\Vert _{L^2}^2+\Vert \nabla u(t,\cdot )\Vert _{L^2}^2\right) \\&\quad \le \frac{1}{2}\int _{ M}\left( |\partial _tu(t,\cdot )|^2+\left( 1-\frac{\rho }{8\pi }\right) |\nabla u(t,\cdot )|^2\right) \\&\quad \le E(u(t,\cdot ))+C\rho =E(u(0,\cdot ))+C\rho , \end{aligned} \end{aligned}$$

where \(C>0\) is independent of \(u(t,\cdot )\).

On the other hand, to estimate \(\Vert u(t,\cdot )\Vert _{L^2}\) we recall that \(\int _{ M}u(t,\cdot )=\int _{ M}u_0\) for all \(t\in [0,T_0)\), see Theorem 1.1, and use the Poincaré inequality to get

$$\begin{aligned} \Vert u(t,\cdot )\Vert _{L^2}&\le \Vert u(t,\cdot )-\overline{u}(t)\Vert _{L^2} + \Vert \overline{u}(t)\Vert _{L^2} \le C\Vert \nabla u(t,\cdot )\Vert _{L^2} + C\overline{u}(t) \\&= C\Vert \nabla u(t,\cdot )\Vert _{L^2} + C\overline{u}_0, \end{aligned}$$

where \(C>0\) is independent of \(u(t,\cdot )\). By the latter estimate and (3.14), we readily have (3.11).

Suppose now one of \(\rho _1,\rho _2\)’s is not positive. Suppose without loss of generality \(\rho _1\le 0\). Then, recalling (3.13) and by using the standard Moser–Trudinger inequality (2.1) we have

$$\begin{aligned} \begin{aligned} E(u(t,\cdot ))&\ge \frac{1}{2}\int _{ M}(|\partial _tu(t,\cdot )|^2+|\nabla u(t,\cdot )|^2)-\rho _2\log \int _{ M}e^{u(t,\cdot )-\overline{u}(t)}\\ &\ge \frac{1}{2}\int _{ M}\left( |\partial _tu(t,\cdot )|^2+\left( 1-\frac{\rho _2}{8\pi }\right) |\nabla u(t,\cdot )|^2\right) -C\rho _2. \end{aligned} \end{aligned}$$

Reasoning as before, one can get (3.11).

Finally, suppose \(\rho _1,\rho _2\le 0\). In this case, we readily have

$$\begin{aligned} E(u(0,\cdot ))=E(u(t,\cdot ))\ge \frac{1}{2}\int _{ M}(|\partial _tu(t,\cdot )|^2+|\nabla u(t,\cdot )|^2), \end{aligned}$$

which yields (3.11). The proof is completed.□

Remark 3.2

For what concerns the wave equation associated with the Toda system (1.7), we can carry out a similar argument to deduce the global existence result in Theorem 1.7. Indeed, for a solution \({\mathbf {u}}=(u_1,\dots ,u_n)\) to (1.7) we define its energy as

$$\begin{aligned} E({\mathbf {u}}(t,\cdot ))= \frac{1}{2}\int _{M}\sum _{i,j=1}^n a^{ij}\left( (\partial _tu_i)(\partial _tu_j) + \langle \nabla u_i,\nabla u_j\rangle \right) -\sum _{i=1}^n\rho _i\log \int _{ M}e^{u_i-\overline{u}_i}, \end{aligned}$$

where \((a^{ij})_{n\times n}\) is the inverse matrix \(A_n^{-1}\) of \(A_n\). Analogous computations as in Lemma 3.1 show that the latter energy is conserved in time, i.e.,

$$\begin{aligned} E({\mathbf {u}}(t,\cdot ))=E({\mathbf {u}}(0,\cdot )) \quad {\text{ for }}\,\,{ \text{ all }}\,\, t\in [0,T]. \end{aligned}$$

To prove the global existence in Theorem 1.7 for \(\rho _i<4\pi\), \(i=1,\dots ,n\), one can then follow the argument of Theorem 1.3 jointly with the Moser–Trudinger inequality associated with the Toda system (1.9), see (2.5).

3.2 Blow-up criteria

We next consider the critical/supercritical case in which \(\rho _i\ge 8\pi\) for some i. The fact that the solutions to (1.2) might blow up makes the problem more delicate. By exploiting the analysis introduced in [6], in particular the improved version of the Moser–Trudinger inequality in Proposition 2.1 and the concentration property in Proposition 2.2, we derive the following general blow-up criteria for (1.1). We stress that this is new for the wave mean field equation (1.4) as well.

Proof of Theorem 1.4

Suppose \(\rho _i\ge 8\pi\) for some i. Let \((u_0,u_1)\in H^1( M)\times L^2( M)\) be such that \(\int _{ M}u_1=0\) and let u be the solution of (1.1) obtained in Theorem 1.1. Suppose that u exists in \([0,T_0)\) for some \(T_0<+\infty\) and it cannot be extended beyond \(T_0\). Then, we claim that there exists a sequence \(t_k\rightarrow T_0^-\) such that either

$$\begin{aligned} \lim _{k\rightarrow \infty }\int _{ M}e^{u(t_k,\cdot )}=+\infty \quad {\text{ or }} \quad \lim _{k\rightarrow \infty }\int _{ M}e^{-u(t_k,\cdot )}=+\infty . \end{aligned}$$
(3.15)

Indeed, suppose this is not the case. Recall the definition of E(u) in (3.9) and the fact that it is conserved in time (3.12). Recall moreover that \(\int _{ M}u(t,\cdot )=\int _{ M}u_0\) for all \(t\in [0,T_0)\), see Theorem 1.1. Then, we would have

$$\begin{aligned}&\frac{1}{2}\int _{ M}(|\partial _tu(t,\cdot )|^2+|\nabla u(t,\cdot )|^2) \\&\quad = E(u(t,\cdot )) +\rho _1\log \int _{ M}e^{u(t,\cdot )-\overline{u}(t)} +\rho _2\log \int _{ M}e^{-u(t,\cdot )+\overline{u}(t)} \\&\quad \le E(u(t,\cdot )) +(\rho _2-\rho _1)\overline{u}(t) +C \\&\quad = E(u(0,\cdot )) +(\rho _2-\rho _1)\overline{u}(0) +C \\&\quad \le C \quad {\text{ for}}\,\,{\text{all}}\,\, t\in [0,T_0), \end{aligned}$$

for some \(C>0\) depending only on \(\rho _1,\rho _2\) and \((u_0,u_1)\). Thus, we can extend the solution u beyond time \(T_0\) contradicting the maximality of \(T_0\). We conclude (3.15) holds true. Now, since \(\overline{u}(t)\) is constant in time, the Moser–Trudinger inequality (2.1) yields

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \nabla u(t_k,\cdot )\Vert _{L^2}=+\infty . \end{aligned}$$
(3.16)

This concludes the first part of Theorem 1.4.

Finally, suppose \(\rho _1\in [8m_1\pi ,8(m_1+1)\pi )\) and \(\rho _2\in [8m_2\pi ,8(m_2+1)\pi )\) for some \(m_1,m_2\in {\mathbb {N}}\), and let \(t_k\) be the above defined sequence. Next, we take \(\tilde{\rho }_i>\rho _i\) such that \(\tilde{\rho }_i\in (8m_i\pi ,8(m_i+1)\pi ),~i=1,2\), and consider the following functional as in (2.3),

$$\begin{aligned} J_{\tilde{\rho }_1,\tilde{\rho }_2}(u)=\frac{1}{2}\int _{ M}|\nabla u|^2-\tilde{\rho }_1\int _{ M}e^{u-\overline{u}}-\tilde{\rho }_2\int _{ M}e^{-u+\overline{u}}. \end{aligned}$$

Since \(\tilde{\rho }_i>\rho _i,~i=1,2\) and since \(E(u(t_i,\cdot ))\), \(\overline{u}(t)\) are preserved in time, we have

$$\begin{aligned} \begin{aligned} J_{\tilde{\rho }_1,\tilde{\rho }_2}(u(t_k,\cdot ))&= E(u(t_k,\cdot ))-\frac{1}{2}\int _{ M}|\partial _tu(t,\cdot )|^2 \\&\quad -(\tilde{\rho }_1-\rho _1)\log \int _{ M}e^{u(t_k,\cdot )-\overline{u}(t_k,\cdot )}-(\tilde{\rho }_2-\rho _2)\log \int _{ M}e^{-u(t_k,\cdot )+\overline{u}(t_k,\cdot )} \\&\le E(u(0,\cdot ))+(\tilde{\rho }_1-\rho _1)\overline{u}(0)-(\tilde{\rho }_2-\rho _2)\overline{u}(0) \\&\quad -(\tilde{\rho }_1-\rho _1)\log \int _{ M}e^{u(t_k,\cdot )}-(\tilde{\rho }_2-\rho _2)\log \int _{ M}e^{-u(t_k,\cdot )}\rightarrow -\infty , \end{aligned} \end{aligned}$$

for \(k\rightarrow +\infty\), where we used (3.15). Then, by the concentration property in Proposition 2.2 applied to the functional \(J_{\tilde{\rho }_1,\tilde{\rho }_2}\), for any \(\varepsilon >0\) we can find either some \(m_1\) points \(\{x_1,\dots ,x_{m_1}\}\subset M\) such that, up to a subsequence,

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _{\cup _{l=1}^{m_1}B_r(x_l)}e^{u(t_k,\cdot )}}{\int _{ M}e^{u(t_k,\cdot )}}\ge 1-\varepsilon , \end{aligned}$$

or some \(m_2\) points \(\{y_1,\dots ,y_{m_2}\}\subset M\) such that

$$\begin{aligned} \lim _{k\rightarrow +\infty }\frac{\int _{\cup _{l=1}^{m_2}B_r(y_l)}e^{-u(t_k,\cdot )}}{\int _{ M}e^{-u(t_k,\cdot )}}\ge 1-\varepsilon . \end{aligned}$$

This finishes the last part of Theorem 1.4.□

Remark 3.3

The general blow-up criteria in Theorem 1.8 for the wave equation associated with the Toda system (1.7) in the critical/super critical regime \(\rho _i\ge 4\pi\) are obtained similarly. More precisely, one has to exploit the conservation of the energy of solutions to (1.7), see Remark 3.2, and the concentration property for the Toda system (1.9) in Proposition 2.3.